Using category theoretic notions, in particular diagrams and their colimits, we provide a common semantic backbone for various notions of modularity in structured ontologies, and outline a general approach for representing (heterogeneous) combinations of ontologies through interfaces of various kinds, based on the theory of institutions. This covers theory interpretations, (definitional) language extensions, symbol identifications, and conservative extensions. In particular, we study the problem of inheriting conservativity between sub-theories in a diagram to its colimit ontology, and apply this to the problem of localisation of reasoning in ‘modular ontology languages’ such as DDLs or E-connections.

Abstract. We propose to use Grothendieck institutions based on 2categorical diagrams as a basis for heterogeneous specification. We prove a number of results about colimits and (some weak variants of) exactness. This framework can also be used for obtaining proof systems for heterogeneous theories involving institution semi-morphisms. 1

We present an overview of the landscape of ontology languages, mostly pertaining to the first-order paradigm. In particular, we present a uniform formalisation of these languages based on the institution theoretical framework, allowing a systematic treatment and analysis of the translational relationships between the various languages and a general analysis of properties of such translations. We also discuss the importance of language translation from the point of view of ontological modularity and logical pluralism, and for the borrowing of tools and reasoners between languages.

Abstract. Several modularity concepts for ontologies have been studied in the literature. Can they be brought to a common basis? We propose to use the language of category theory, in particular diagrams and their colimits, for answering this question. We outline a general approach for representing combinations of logical theories, or ontologies, through interfaces of various kinds, based on diagrams and the theory of institutions. In particular, we consider theory interpretations, language extensions, symbol identification, and conservative extensions. We study the problem of inheriting conservativity between sub-theories in a diagram to its colimit ontology. Finally, we apply this to the problem of conservativity when composing DDLs or E-connections. 1

Abstract. We show that any institution I satisfying some reasonable conditions can be transformed into another institution, Ibeh, which captures formally and abstractly the intuitions of adding support for behavioral equivalence and reasoning to an existing, particular algebraic framework. We call our transformation an “extension ” because Ibeh has the same sentences as I and because its entailment relation includes that of I. Many properties of behavioral equivalence in concrete hidden logics follow as special cases of corresponding institutional results. As expected, the presented constructions and results can be instantiated to other logics satisfying our requirements as well, thus leading to novel behavioral logics, such as partial or infinitary ones, that have the desired properties. 1

We introduce a semantic encoding of partial algebras as total algebras through a Horn axiomatization of the existence equality relation interpreted as an algebraic operation. We show that this novel encoding enjoys several important properties that make it a good tool for the execution of partial algebraic specifications through means specific to ordinary algebraic reasoning, such as term rewriting.

The Semantic Web (SW) is viewed as the next generation of the Web that enables intelligent software agents to process and aggregate data autonomously. Ontology languages provide basic vocabularies to semantically markup data on the SW. We have witnessed an increase of numbers of SW languages in the last years. These languages, such as RDF, RDF Schema (RDFS), the OWL suite of languages, the OWL − suite, SWRL, are based on different semantics, such as the RDFS-based, description logic-based, Datalog-based semantics. The relationship among the various semantics poses a challenge for the SW community for making the languages interoperable. Institutions provide a means of reasoning about software specifications regardless of the logical system. This makes it an ideal candidate to represent and reason about the various languages in the Semantic Web. In this paper, we construct institutions for the SW languages and use institution morphisms to relate them. We show that RDF framework together with the RDF serializations of SW languages form an indexed institution. This allows the use of Grothendieck institutions to combine Web ontologies described in various languages.

This paper systematically applies tools and techniques from the area of algebraic specification theory to corresponding ontology structuring and design tasks. We employ the heterogeneous structuring mechanisms of the heterogeneous algebraic specification language HetCasl for defining an abstract notion of structured heterogeneous ontology. This approach enables the designer to split up a heterogeneous ontology into semantically meaningful parts and employ dedicated reasoning tools to them. In particular, we distinguish three fundamentally different kinds of combining heterogeneous ontologies: integration, connection, and refinement.

Abstract. We present a general approach for representing, composing, and computing alignments, based on the category theoretic notions of diagram, pushout, and colimit. This generalises the possible ‘shapes ’ of alignments that have been introduced previously in similar approaches. We use the theory of institutions to represent heterogeneous ontologies, and show how the tool Hets can be employed to compute the colimit ontology of an alignment diagram. 1

Heterogeneous multi-logic theories arise in different contexts: they are needed for the specification of large software systems, as well as for mediating between different